Question

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vector and $\theta$ is the angle between them. Then $\overrightarrow{a}+\overrightarrow{b}$ is a unit vector if

• A. $\theta =\frac{\pi }{4}$
• B. $\theta =\frac{\pi }{3}$
• C. $\theta =\frac{\pi }{2}$
• D. $\theta =\frac{2\pi }{3}$

Answer 1

The correct option is D $\theta =\frac{2\pi }{3}$

$\overline{a}+\overline{b}$ is a unit vector if $|\overline{a}+\overline{b}|=1$

$|a+b{|}^2=1\Rightarrow (\overline{a}+\overline{b}).(\overline{a}+\overline{b})=1$

$\Rightarrow |a{|}^2+|b{|}^2+2(\overline{a}.\overline{b})=1$

$\Rightarrow 1+1+2(\overline{a}.\overline{b})=1\therefore 2(\overline{a}.\overline{b})=-1$

$\Rightarrow \overline{a}.\overline{b}=\frac{-1}{2}$

$\Rightarrow |a||\overline{b}|cos\theta =\frac{-1}{2}$

$\Rightarrow 1\times 1\times cos\theta =\frac{-1}{2}\Rightarrow cos\theta =\frac{-1}{2}$

$\theta =\frac{2\pi }{3},$ hence answer is $\left(D\right)$